Now, given a process \(x\)  which has RCLL paths, \(x_{0-}:=0\)  and \(\Delta x_t:=x_t-x_{t-},t\ge 0\) , let \(m\)  denote the measure on \(\mathcal {F}\otimes \mathcal {B}([0,T])\otimes \mathcal {B}(\mathcal {E})\)  generated by \(N\)  that \(m(A)=\bar{E}\int _0^T\int _{\mathcal {E}}\mathbb {I}_{A}N(de,dt)\) . For any \(\mathcal {F}\otimes \mathcal {B}([0,T])\otimes \mathcal {B}(\mathcal {E})/\mathcal {B}(\mathbb {R})\)  measurable integrable process \(x\) , we set \(\mathbb {E}[x]:=\int xdm\)  and denote by \(\mathbb {E}[x|\mathcal {P}\otimes \mathcal {B}(\mathcal {E})]\)  the Radon-Nikodym derivatives w.r.t. \(\mathcal {P}\otimes \mathcal {B}(\mathcal {E})\) . In fact, \(\mathbb {E}\)  is not an expectation (for \(m\)  is not a probability measure), but it owns similar properties to expectation. A more general definition of stochastic integral of random measure has been introduced by [1]} where the theory of dual predictable projection is utilized. Therefore, we omit the details here and give the following lemma directly.
